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I have this question:

Write the linear system $$\begin{array}{rcr}-2x_1+x_2-4x_3 & = & 1 \\ x_1-2x_2 & = & -3 \\ x_1+x_2-4x_3 & = & 0 \end{array}$$ in the matrix-vector form $A\mathbf{x}=\mathbf{b}$.

Is this what they want?

$$ x_1* \begin{bmatrix} -2 \\ 1 \\ 1 \end{bmatrix} + x_2* \begin{bmatrix} 1 \\ -2 \\ 1 \end{bmatrix} + x_3* \begin{bmatrix} -4 \\ 0 \\ -4 \end{bmatrix} = \begin{bmatrix} 1 \\ -3 \\ 0 \end{bmatrix} $$

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2 Answers

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I guess the matrix-vector form here refers to the matrix A and the vector b. I would suggest to rewrite the equation in the following way

$$\begin{pmatrix}-2&1&-4\\1&-2&0\\1&1&-4\end{pmatrix}\begin{pmatrix}x_1\\x_2\\x_3\end{pmatrix}~=~\begin{pmatrix}1\\-3\\0\end{pmatrix}$$

To verify the L.H.S. you can just multiply the vector by the matrix and then your will get first guess.

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basically!

$ A = \begin{pmatrix} -2 & 1 & -4 \\ 1 & -2 & 0 \\ 1 & 1 & -4 \end{pmatrix} $ and $b = \begin{pmatrix} 1 \\ -3 \\ 0 \end{pmatrix}$ yielding $Ax = b$.

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